The above are sori, found on the underside of a 'true' fern frond. In each of the sori are spores which are haploid (have only half of the chromosomes as the adult stage). These spores are released into puddles of water, where they create a structure called a prothallus. This is a heart shaped structure, which is only a cell or two thick and fairly small. The prothallus produces either eggs, sperm, or sometimes both (depending on the species). The sperm have to swim to a different prothallus to fertilize an egg, as the eggs are retained there. The fertilized egg is what becomes the fern, and it grows out of the prothallus.
So the thin tissue that the small leaves are growing out of is the prothallus, and the leaves are the young fern. Later leaves will look more fern-like (shape depends on the species) and grow in the characteristic fiddle-head shape.